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Mirrors > Home > MPE Home > Th. List > ereldm | Structured version Visualization version GIF version |
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereldm.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ereldm.2 | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Ref | Expression |
---|---|
ereldm | ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ereldm.2 | . . . 4 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | |
2 | 1 | neeq1d 3004 | . . 3 ⊢ (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅)) |
3 | ecdmn0 8696 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) | |
4 | ecdmn0 8696 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅)) |
6 | ereldm.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
7 | erdm 8659 | . . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 8 | eleq2d 2824 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋)) |
10 | 8 | eleq2d 2824 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋)) |
11 | 5, 9, 10 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∅c0 4283 dom cdm 5634 Er wer 8646 [cec 8647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-er 8649 df-ec 8651 |
This theorem is referenced by: erth 8698 brecop 8750 eceqoveq 8762 |
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